# P-adic methods in Automorphic Forms

On Wednesday, July 27th 2022, we will host an informal one-day workshop with the loose theme 'p-adic methods in automorphic forms', with three external talks.

###### Practicalities

The talks will all take place in B3.02 of the Zeeman building, and be broadcast on the Warwick Number Theory team on MS Teams (in the seminar channel). We do not post this link publicly, but are happy to share the link privately to any who would like to follow the talks remotely.

There are some funds to support the travel costs of participants. If you'd like to attend, please email Chris at christopher.d.williams@warwick.ac.uk.

###### Schedule

11.00 - 12.00 | Daniel Barrera (Santiago) |

12.00 - 13.30 | Lunch |

13.30 - 14.30 | Andrew Graham (Paris) |

14.30 - 15.15 | Coffee |

15.15 - 16.15 | Sadiah Zahoor (Sheffield) |

**Titles and abstracts**

**Daniel Barrera: Branching laws and p-adic deformation**

In this talk we will try to explain the role played by branching laws in the construction and deformation of objects which are relevant in arithmetic. More precisely, firstly we will consider p-adic deformation of $p$-adic $L$-functions for via the use of branching laws. Secondly, in the context of GL(2) x GL(2) x GL(2) over totally real number fields we will explain their use to study Galois cohomology classes.

**Andrew Graham: Locally analytic actions on nearly overconvergent modular forms**

Recent work of Howe shows that the action of the Atkin--Serre operator on $p$-adic modular forms can be reinterpreted as a $\widehat{\mathbb{G}}_m$ action on the Katz Igusa tower. By $p$-adic Fourier theory, this gives an action of continuous functions on $\mathbb{Z}_p$ on sections of the Igusa tower ($p$-adic modular forms). We show that this action ``overconverges'' in the sense that the subspace of nearly overconvergent modular forms is stable under the action of locally analytic functions on $\mathbb{Z}_p$. This recovers (but is more general than) the construction of Andreatta--Iovita and has applications to the construction of $p$-adic $L$-functions. This is joint work with Vincent Pilloni and Joaquin Rodrigues.

**Sadiah Zahoor: ****Congruences for half-integral weight modular forms**

The theory of half-integral weight modular forms may be adopted to prove 'congruences' between Selmer type groups. In the paper 'Modular Form Congruences and Selmer Groups', McGraw and Ono used this approach by passing on to 'congruences' between modular forms of integer and half-integer weight. Recall the famous 'congruence modulo 11 between the normalised Discriminant function Delta of weight 12 and the newform f of weight 2 attached to Elliptic curve of conductor 11. Using Shimura's correspondence (1973), which connects modular forms of integer weight 2k with half-integer modular forms of weight k+(1/2), our congruence descends to a congruence modulo 11 between half integer modular forms of weight (3/2) and (13/2). We show such congruences hold in general.

The talk shall begin with a brief introduction to modular forms of integer and half-integer weight leading to congruences modulo an odd prime p between them. I will also give an overview of current progress and generalisation of Theorem of McGraw and Ono to Hilbert modular forms of integer and half-integer weight.